Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
197 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137 | C2 | semidirect |
| 3. C2×#124 | C2 | semidirect |
| 4. C2×#104 | C2 | semidirect |
| 5. #419 | C2 | semidirect |
| 6. C2×#100 | C2 | semidirect |
| 7. #339 | C2 | semidirect |
| 40. C2×Q8 | D4 | semidirect |
| 59. M16 | D4 | semidirect |
| 68. C42 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| #535 | 2 |
| C2×#124 | 2 |
| C2×#104 | 2 |
| #419 | 2 |
| C2×#100 | 2 |
| #339 | 2 |